Average Error: 0.2 → 0.1
Time: 16.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x\right) + 1}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x\right) + 1}{x - 1}}
double f(double x) {
        double r681345 = 6.0;
        double r681346 = x;
        double r681347 = 1.0;
        double r681348 = r681346 - r681347;
        double r681349 = r681345 * r681348;
        double r681350 = r681346 + r681347;
        double r681351 = 4.0;
        double r681352 = sqrt(r681346);
        double r681353 = r681351 * r681352;
        double r681354 = r681350 + r681353;
        double r681355 = r681349 / r681354;
        return r681355;
}
↓
double f(double x) {
        double r681356 = 6.0;
        double r681357 = x;
        double r681358 = sqrt(r681357);
        double r681359 = 4.0;
        double r681360 = fma(r681358, r681359, r681357);
        double r681361 = 1.0;
        double r681362 = r681360 + r681361;
        double r681363 = r681357 - r681361;
        double r681364 = r681362 / r681363;
        double r681365 = r681356 / r681364;
        return r681365;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \color{blue}{\left(1 \cdot 6\right)} \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]

  5. Applied associate-*l*0.0

    \[\leadsto \color{blue}{1 \cdot \left(6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right)}\]

  6. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x\right) + 1}{x - 1}}}\]

  7. Final simplification0.1

    \[\leadsto \frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x\right) + 1}{x - 1}}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))